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Simplifying (50 + -2x)(40 + -2x) = 1000 Multiply (50 + -2x) * (40 + -2x) (50(40 + -2x) + -2x * (40 + -2x)) = 1000 ((40 * 50 + -2x * 50) + -2x * (40 + -2x)) = 1000 ((2000 + -100x) + -2x * (40 + -2x)) = 1000 (2000 + -100x + (40 * -2x + -2x * -2x)) = 1000 (2000 + -100x + (-80x + 4x2)) = 1000 Combine like terms: -100x + -80x = -180x (2000 + -180x + 4x2) = 1000 Solving 2000 + -180x + 4x2 = 1000 Solving for variable 'x'. Reorder the terms: 2000 + -1000 + -180x + 4x2 = 1000 + -1000 Combine like terms: 2000 + -1000 = 1000 1000 + -180x + 4x2 = 1000 + -1000 Combine like terms: 1000 + -1000 = 0 1000 + -180x + 4x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(250 + -45x + x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(250 + -45x + x2)' equal to zero and attempt to solve: Simplifying 250 + -45x + x2 = 0 Solving 250 + -45x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '-250' to each side of the equation. 250 + -45x + -250 + x2 = 0 + -250 Reorder the terms: 250 + -250 + -45x + x2 = 0 + -250 Combine like terms: 250 + -250 = 0 0 + -45x + x2 = 0 + -250 -45x + x2 = 0 + -250 Combine like terms: 0 + -250 = -250 -45x + x2 = -250 The x term is -45x. Take half its coefficient (-22.5). Square it (506.25) and add it to both sides. Add '506.25' to each side of the equation. -45x + 506.25 + x2 = -250 + 506.25 Reorder the terms: 506.25 + -45x + x2 = -250 + 506.25 Combine like terms: -250 + 506.25 = 256.25 506.25 + -45x + x2 = 256.25 Factor a perfect square on the left side: (x + -22.5)(x + -22.5) = 256.25 Calculate the square root of the right side: 16.007810594 Break this problem into two subproblems by setting (x + -22.5) equal to 16.007810594 and -16.007810594.Subproblem 1
x + -22.5 = 16.007810594 Simplifying x + -22.5 = 16.007810594 Reorder the terms: -22.5 + x = 16.007810594 Solving -22.5 + x = 16.007810594 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '22.5' to each side of the equation. -22.5 + 22.5 + x = 16.007810594 + 22.5 Combine like terms: -22.5 + 22.5 = 0.0 0.0 + x = 16.007810594 + 22.5 x = 16.007810594 + 22.5 Combine like terms: 16.007810594 + 22.5 = 38.507810594 x = 38.507810594 Simplifying x = 38.507810594Subproblem 2
x + -22.5 = -16.007810594 Simplifying x + -22.5 = -16.007810594 Reorder the terms: -22.5 + x = -16.007810594 Solving -22.5 + x = -16.007810594 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '22.5' to each side of the equation. -22.5 + 22.5 + x = -16.007810594 + 22.5 Combine like terms: -22.5 + 22.5 = 0.0 0.0 + x = -16.007810594 + 22.5 x = -16.007810594 + 22.5 Combine like terms: -16.007810594 + 22.5 = 6.492189406 x = 6.492189406 Simplifying x = 6.492189406Solution
The solution to the problem is based on the solutions from the subproblems. x = {38.507810594, 6.492189406}Solution
x = {38.507810594, 6.492189406}
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